screen composition in blender
You can use the compositor for this.

Normally I add 2 materials to the object. One for the case and one for the screen. The faces for the screen I assign to the screen material.

When the screen material is selected under settings you can set the pass index to 1 (default is 0). With this pass index you can extract the visible parts in the compositor



for the screen make sure that the UV coordinates are correct.
make sure you select "material index" and "UV" in the render passes

In the compositor you can then extract the mask and apply a different texture using the UV Map node


Normally in these cases you can render out to OpenEXR Multilayer, and load the prerendered images directly in the compositor.